Find The 7 Th 7^{\text{th}} 7 Th Term In The Sequence: − 4 , − 8 , − 16 , … -4, -8, -16, \ldots − 4 , − 8 , − 16 , … Hint: Write A Formula To Help You. Use The Formula: 1st term × ( Common Ratio ) ( Desired term − 1 ) \text{1st Term} \times (\text{Common Ratio})^{(\text{desired Term} - 1)} 1st term × ( Common Ratio ) ( Desired term − 1 )
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Understanding Geometric Sequences
A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The formula to find the nth term of a geometric sequence is given by:
a_n = a_1 * r^(n-1)
where:
- a_n is the nth term of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the desired term number
Identifying the First Term and Common Ratio
In the given sequence: -4, -8, -16, ..., we can see that each term is obtained by multiplying the previous term by -2. Therefore, the common ratio (r) is -2.
The first term (a_1) is -4.
Applying the Formula to Find the 7th Term
Now that we have the first term and the common ratio, we can use the formula to find the 7th term:
a_7 = a_1 * r^(7-1) a_7 = -4 * (-2)^(7-1) a_7 = -4 * (-2)^6 a_7 = -4 * 64 a_7 = -256
Therefore, the 7th term in the sequence is -256.
Why the Formula Works
The formula works because each term in a geometric sequence is obtained by multiplying the previous term by the common ratio. By raising the common ratio to the power of (n-1), we are essentially multiplying the first term by the common ratio (n-1) times. This gives us the nth term of the sequence.
Real-World Applications of Geometric Sequences
Geometric sequences have many real-world applications, such as:
- Population growth: The population of a city or country can be modeled using a geometric sequence, where each term represents the population at a given time.
- Financial investments: The value of an investment can be modeled using a geometric sequence, where each term represents the value of the investment at a given time.
- Physics: The motion of an object can be modeled using a geometric sequence, where each term represents the position of the object at a given time.
Conclusion
In conclusion, finding the 7th term in a geometric sequence involves identifying the first term and the common ratio, and then applying the formula to find the desired term. The formula works because each term in a geometric sequence is obtained by multiplying the previous term by the common ratio. Geometric sequences have many real-world applications, and understanding how to find the nth term of a geometric sequence is an important skill in mathematics and other fields.
Example Problems
- Find the 5th term in the sequence: 2, 6, 18, ...
- Find the 3rd term in the sequence: 3, -9, 27, ...
- Find the 2nd term in the sequence: 4, -16, 64, ...
Solutions
- a_5 = a_1 * r^(5-1) a_5 = 2 * 3^(5-1) a_5 = 2 * 3^4 a_5 = 2 * 81 a_5 = 162
- a_3 = a_1 * r^(3-1) a_3 = 3 * (-3)^(3-1) a_3 = 3 * (-3)^2 a_3 = 3 * 9 a_3 = 27
- a_2 = a_1 * r^(2-1) a_2 = 4 * (-4)^(2-1) a_2 = 4 * (-4)^1 a_2 = 4 * -4 a_2 = -16
Practice Problems
- Find the 4th term in the sequence: 1, -3, 9, ...
- Find the 6th term in the sequence: 2, 6, 18, ...
- Find the 1st term in the sequence: 3, -9, 27, ...
Solutions
- a_4 = a_1 * r^(4-1) a_4 = 1 * (-3)^(4-1) a_4 = 1 * (-3)^3 a_4 = 1 * -27 a_4 = -27
- a_6 = a_1 * r^(6-1) a_6 = 2 * 3^(6-1) a_6 = 2 * 3^5 a_6 = 2 * 243 a_6 = 486
- a_1 = a_1 a_1 = 3
Tips and Tricks
- Make sure to identify the first term and the common ratio before applying the formula.
- Use the formula to find the desired term, and then simplify the expression to get the final answer.
- Practice, practice, practice! The more you practice finding the nth term of a geometric sequence, the more comfortable you will become with the formula and the more confident you will be in your ability to solve problems.
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Q: What is a geometric sequence?
A: A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Q: How do I find the nth term of a geometric sequence?
A: To find the nth term of a geometric sequence, you can use the formula:
a_n = a_1 * r^(n-1)
where:
- a_n is the nth term of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the desired term number
Q: What is the common ratio?
A: The common ratio (r) is the number by which each term in a geometric sequence is multiplied to get the next term.
Q: How do I find the common ratio?
A: To find the common ratio, you can divide any term by its previous term. For example, if the sequence is 2, 6, 18, ..., you can divide 6 by 2 to get 3, or divide 18 by 6 to get 3.
Q: What is the first term?
A: The first term (a_1) is the first term in the sequence.
Q: How do I find the first term?
A: To find the first term, you can look at the sequence and identify the first number. For example, if the sequence is 2, 6, 18, ..., the first term is 2.
Q: Can I use the formula to find any term in a geometric sequence?
A: Yes, you can use the formula to find any term in a geometric sequence, as long as you know the first term and the common ratio.
Q: What if I don't know the first term or the common ratio?
A: If you don't know the first term or the common ratio, you may need to use other methods to find the term, such as using a table or a graph.
Q: Can I use the formula to find the sum of a geometric sequence?
A: Yes, you can use the formula to find the sum of a geometric sequence. The formula for the sum of a geometric sequence is:
S_n = a_1 * (1 - r^n) / (1 - r)
where:
- S_n is the sum of the first n terms of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the number of terms
Q: What if the common ratio is 1?
A: If the common ratio is 1, the sequence is not geometric, but rather arithmetic. In this case, you can use the formula for the sum of an arithmetic sequence:
S_n = n/2 * (a_1 + a_n)
where:
- S_n is the sum of the first n terms of the sequence
- a_1 is the first term of the sequence
- a_n is the nth term of the sequence
- n is the number of terms
Q: What if the common ratio is -1?
A: If the common ratio is -1, the sequence alternates between positive and negative terms. In this case, you can use the formula for the sum of a geometric sequence:
S_n = a_1 * (1 - r^n) / (1 - r)
where:
- S_n is the sum of the first n terms of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the number of terms
Q: Can I use the formula to find the product of a geometric sequence?
A: Yes, you can use the formula to find the product of a geometric sequence. The formula for the product of a geometric sequence is:
P_n = a_1 * r^(n-1)
where:
- P_n is the product of the first n terms of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the number of terms
Q: What if I want to find the nth term of a geometric sequence with a negative common ratio?
A: To find the nth term of a geometric sequence with a negative common ratio, you can use the formula:
a_n = a_1 * (-r)^(n-1)
where:
- a_n is the nth term of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the desired term number
Q: Can I use the formula to find the sum of a geometric sequence with a negative common ratio?
A: Yes, you can use the formula to find the sum of a geometric sequence with a negative common ratio. The formula for the sum of a geometric sequence with a negative common ratio is:
S_n = a_1 * (1 - (-r)^n) / (1 - (-r))
where:
- S_n is the sum of the first n terms of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the number of terms
Q: What if I want to find the product of a geometric sequence with a negative common ratio?
A: To find the product of a geometric sequence with a negative common ratio, you can use the formula:
P_n = a_1 * (-r)^(n-1)
where:
- P_n is the product of the first n terms of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the number of terms
Q: Can I use the formula to find the sum of a geometric sequence with a common ratio of 0?
A: No, you cannot use the formula to find the sum of a geometric sequence with a common ratio of 0. In this case, the sequence is not geometric, but rather constant. The sum of a constant sequence is simply the first term multiplied by the number of terms.
Q: What if I want to find the product of a geometric sequence with a common ratio of 0?
A: To find the product of a geometric sequence with a common ratio of 0, you can use the formula:
P_n = a_1
where:
- P_n is the product of the first n terms of the sequence
- a_1 is the first term of the sequence
- n is the number of terms
Q: Can I use the formula to find the sum of a geometric sequence with a common ratio of 1?
A: No, you cannot use the formula to find the sum of a geometric sequence with a common ratio of 1. In this case, the sequence is not geometric, but rather arithmetic. You can use the formula for the sum of an arithmetic sequence:
S_n = n/2 * (a_1 + a_n)
where:
- S_n is the sum of the first n terms of the sequence
- a_1 is the first term of the sequence
- a_n is the nth term of the sequence
- n is the number of terms
Q: What if I want to find the product of a geometric sequence with a common ratio of 1?
A: To find the product of a geometric sequence with a common ratio of 1, you can use the formula:
P_n = a_1 * r^(n-1)
where:
- P_n is the product of the first n terms of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the number of terms
Q: Can I use the formula to find the sum of a geometric sequence with a common ratio of -1?
A: Yes, you can use the formula to find the sum of a geometric sequence with a common ratio of -1. The formula for the sum of a geometric sequence with a common ratio of -1 is:
S_n = a_1 * (1 - (-1)^n) / (1 - (-1))
where:
- S_n is the sum of the first n terms of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the number of terms
Q: What if I want to find the product of a geometric sequence with a common ratio of -1?
A: To find the product of a geometric sequence with a common ratio of -1, you can use the formula:
P_n = a_1 * (-r)^(n-1)
where:
- P_n is the product of the first n terms of the sequence
- a_1 is the first term of the sequence
- r is the